Question

16. Which of the following is the recurrence relation for the power series solution about x=0 of the given equation? (8 Puan) y"-2xy + 4By = 0 where is a constant 2(-28 2n+2 = (m+2)(n+1)+1 none of these an+2= 2(n+B) an (n+2)(n+3) O an+2 2(-28) (n+2)(n+1) a Ant2 = 2() (7+2)(n+1) co 206-) m+2 (+1+2)(n+1) 2 - (+2)(n+11 an 20+) a 272)n+3) C+!

Answer 1

Put these value & in 7 Anss y" - gx 4' +4BY = 0 Let suppose around o we have power series solution 4= Ě Onxn nzo n-1 y = E nanan- n=1 y! me E n(n-1) Anxh-2 n=2 n- E n(n-1) anx 2 x Ě hanxhi +4B E Chun=0 n=o n=2 nel + 4B Anxh= =0 En(n-1) an xn-2 g nana no n=2 for a En(n-1) anx"-2 Let n=k+2 n=2 xk E n(n-1) an XX-2 - E (K+2) (K+1) Akt2 K=o n=2 so newrite n-2 E b/n-1) Anx E Ē Int2)(n+1) Antex an n=2 nzo

Now from ** 2. E (2+2)(n+1) Ontz xn-2 & hand" + 4B n=1 20₂ + E (+2)(n+1) On+ 2 X 2 Enanth + 4 Blot Eanx² = 0 n=1 X4B nal h=) 20, +4B 00 + Eſin+2)(n+1) An+2 - 20n n + en pen=0 4B n=1 By comparison 43 (n+2)(n+1) Anta 2 anin tanto Anth = an an-Anx4 B int? 1/n+1) 2 an(n-2B) inti )(n+2) Anta Ans option (4) is right
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